Download Algebra note sheet test 10

Algebra note sheet test 15
Block 1
Solving systems of equations:
Using the elimination method:
1) Get the coefficient on either the x or the y to be the same number with opposite signs.
2) Add the two equations together to eliminate one of the variables.
3) Solve for the remaining variable
4) Plug back into either equation to solve for the other variable. Your answer is an ordered pair.
Using the substitution method:
1) Isolate a variable in either equation
2) Plug the equivalent expression that the variable is equal to into the other equation.
3) Solve the new equation for the variable.
4) Plug that answer into the isolated equation to find the value of the other variable. Your answer
with be an ordered pair.
consistent – a system that has a least one solution
inconsistent – a system with no solution
independent – a system with exactly one solution
dependent – a system with infinitely many solutions
Block 2
Transformation of linear graphs:
3 transformations
1) changing the “b” value shifts the graph up or down. Adding shifts up, subtracting shifts down
This is a translation or slide.
2) changing the “m” value or slope, changes the steepness of the graph. Values of “m” that are
between 0 & 1 will flatten the line or make it less steep. Values of “m” that are more than 1
will make the graph steeper. This is a rotation about a point.
3) if the “m” is a negative value, the line flips over the x-axis and all the above is also true regarding the
size of the “m” value. This is a reflection.
Solving systems of equations:
To determine if an ordered pair is a solution to a system of equations just plug the point into each equation
and determine if it is a solution for each.
To find a solution to a system of equation by graphing, simply graph each line and find the point of intersection.
Block 3
Finding the line of best fit and the sum of the squares of the residuals:
This is an extra credit problem but all you need to do it find out how far the y-value s for each equation
are from the given y-values. Once you determine how far the y-values are from the given y-values, you
square their differences and add them up. The line with the least sum of squares is the line of best fit
3 Forms of equations and their slopes:
1) Ax+By=C This is standard form and it has a slope that is m =
2)
=m(x- ) This is point-slope form and the slope is contained within the equation. It is the “m”.
3) y=mx+b This is slope-intercept form and the slope is also contained in the equations. It is the “m”.
Parallel lines have slopes that are equal
Perpendicular lines have slopes that have opposite signs and are reciprocals of each other.
To determine if two equations are going to have parallel or perpendicular lines, find the slopes and compare.
Writing equations that are parallel or perpendicular to a given line with a given point:
First, determine what the slope of the given line is. For parallel lines, use that slope and the given
point to write the equation of the line using one of the two methods used in Block 5.
For perpendicular lines, take the reciprocal of the slope you found and change the sign, then plug in the
given point to write the equation of the line using one of the two methods used in Block 5.
Block 4
Finding the slope given two points:
Slope wi11 always be a fraction where the change in the “y” values is the top number and the change in the
“x” values is the bottom number. You can also use the slope formula: m 
y 2  y1
x 2  x1
To find the slope of a line in standard form (Ax+By=C):
m
A
B
Direct Variation:
To determine if a relation is a direct variation, pick two points and plug the x-coordinates and the
y-coordinates into the proportion
x1 x 2

and see if the cross-products are equal, if they are the
y1 y 2
relation is a direct variation in the form y = kx.
Graphing a line give the slope and y-intercept:
Plot the y-intercept and then use the slope to find a second point, then just connect the y-intercept and the
second point.
Block 5
Point-slope form
where “m” is the slope and
and
give you a given point (
Slope-intercept form
y = mx + b where “m” is the slope and “b” is the y-intercept
To write an equation in slope-intercept form from a given point and a slope, you have two choices.
1) plug the slope and a point into point-slope form, then distribute the slope through the parenthesis and
eliminate the number on the same side as the “y”. (i.e. – get the “y” by itself)
2) plug the “x” the “y” and the “m” into y = mx + b and solve for “b”. Then just plug “m” and “b” into
y = mx + b
To write and equation in slope-intercept form from standard form:
Manipulate the equation until you get the “y” by itself.
To graph an equation in point-slope form, pull out the slope and the given point. Plot the point and use the slope
to get a second point, then connect the two points.